Algebra Examples

$\frac{1}{3} \leq \frac{x + 1}{4} < \frac{4}{5}$

Step 1

Solve $\frac{1}{3} \leq \frac{x + 1}{4}$ for $x$ .

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Step 1.1

Multiply both sides by $4$ .

$\frac{1}{3} \cdot 4 \leq \frac{x + 1}{4} \cdot 4$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Combine $\frac{1}{3}$ and $4$ .

$\frac{4}{3} \leq \frac{x + 1}{4} \cdot 4$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{4}{3} \leq \frac{x + 1}{4} \cdot 4$

Step 1.2.2.1.2

Rewrite the expression.

$\frac{4}{3} \leq x + 1$

Step 1.3

Solve for $x$ .

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Step 1.3.1

Rewrite so $x$ is on the left side of the inequality.

$x + 1 \geq \frac{4}{3}$

Step 1.3.2

Move all terms not containing $x$ to the right side of the inequality.

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Step 1.3.2.1

Subtract $1$ from both sides of the inequality.

$x \geq \frac{4}{3} - 1$

Step 1.3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x \geq \frac{4}{3} - 1 \cdot \frac{3}{3}$

Step 1.3.2.3

Combine $- 1$ and $\frac{3}{3}$ .

$x \geq \frac{4}{3} + \frac{- 1 \cdot 3}{3}$

Step 1.3.2.4

Combine the numerators over the common denominator.

$x \geq \frac{4 - 1 \cdot 3}{3}$

Step 1.3.2.5

Simplify the numerator.

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Step 1.3.2.5.1

Multiply $- 1$ by $3$ .

$x \geq \frac{4 - 3}{3}$

Step 1.3.2.5.2

Subtract $3$ from $4$ .

$x \geq \frac{1}{3}$

Step 2

Solve $\frac{x + 1}{4} < \frac{4}{5}$ for $x$ .

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Step 2.1

Multiply both sides by $4$ .

$\frac{x + 1}{4} \cdot 4 < \frac{4}{5} \cdot 4$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x + 1}{4} \cdot 4 < \frac{4}{5} \cdot 4$

Step 2.2.1.1.2

Rewrite the expression.

$x + 1 < \frac{4}{5} \cdot 4$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $\frac{4}{5} \cdot 4$ .

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Step 2.2.2.1.1

Combine $\frac{4}{5}$ and $4$ .

$x + 1 < \frac{4 \cdot 4}{5}$

Step 2.2.2.1.2

Multiply $4$ by $4$ .

$x + 1 < \frac{16}{5}$

Step 2.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.3.1

Subtract $1$ from both sides of the inequality.

$x < \frac{16}{5} - 1$

Step 2.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x < \frac{16}{5} - 1 \cdot \frac{5}{5}$

Step 2.3.3

Combine $- 1$ and $\frac{5}{5}$ .

$x < \frac{16}{5} + \frac{- 1 \cdot 5}{5}$

Step 2.3.4

Combine the numerators over the common denominator.

$x < \frac{16 - 1 \cdot 5}{5}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $- 1$ by $5$ .

$x < \frac{16 - 5}{5}$

Step 2.3.5.2

Subtract $5$ from $16$ .

$x < \frac{11}{5}$

Step 3

Find the intersection of $x \geq \frac{1}{3}$ and $x < \frac{11}{5}$ .

$\frac{1}{3} \leq x < \frac{11}{5}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$\frac{1}{3} \leq x < \frac{11}{5}$

Interval Notation:

$[\frac{1}{3}, \frac{11}{5})$

Step 5